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This entry has nothing to do with numerical methods, mathematics, logic, gaming, or anything else I've written about. Well, there may be a few numbers. Maybe. One day in January this year, while spending precious minutes of my life browsing YouTube as people tend to do on occasion, my clicks led from one thing to another ranging from music videos to Maker projects to magic acts, and then I landed on a 9-minute video called "How Penn Jillette Lost over 100 Lbs and Still Eats Whatever He Wants," featuring Las Vegas magician Penn Jillette talking about his experience with what seemed to be a dangerously unhealthy diet in which he ate nothing but potatoes for two weeks. It's called a monotrophic diet or simply mono diet. The idea is, you eat just one food ingredient (like potatoes) for a while, and you lose weight due to the calorie deficit. I mean, who wouldn't lose weight if you're hungry and you know that your only option is to eat another potato? You can tole…

My son and I began playing Minecraft together, me on my Windows laptop and him on his iPad. In my opinion, Minecraft is probably the only game in which a parent and child can play together in the same virtual world and both of them enjoy it. Quality time with your child in a video game? Who would have thought?The game is easy enough for a child to understand and complex enough for an adult to appreciate. In "creative" mode, it's like a giant Lego virtual 3D world in which you can build fantastic structures more easily than with Lego. Then there's "survival" mode, in which you're plunked down in a world with nothing, and you must gather resources, make your own tools, feed yourself, and so on — pulling yourself up by your bootstraps in a world full of both friendly and hostile creatures.One of the more intriguing features of Minecraft is the redstone circuit capability. You can build moving machines, logic circuits, devices that get triggered from a foot…

So now we're ready to put all the calculations together: launch tube, water thrust, air thrust, and ballistic flight. The launch tube calculation is more like an initialization for the water thrust phase; we just have one time step of the duration of launch tube traversal. Get the initial pressure \(P_0\) for water thrust from (L1), and the velocity \(v_0\) from (L4), and use these values in the first water thrust time step. You may also want to initialize the altitude achieved as the height of the tube. Water thrust calculationFor the water thrust phase, we do the following calculations at each time interval, which should be 1 millisecond or less in duration:At the beginning of time interval \(i\):Get the pressure \(P_1\) from (W4), using the change in water volume from the previous time interval.From the water volume, calculate the height of the water in the bottle — needed for \(H\) in equation (W1). This can be approximated by calculating the height of the water in a cylinder …

After the water is expended, the tank (bottle) still contains substantial pressure that releases quickly through the nozzle. This final burst of air can impart a significant boost to velocity (at least 30% depending on the mass of the rocket), so we shouldn’t ignore this contribution to thrust. For convenience, however, we will ignore any further affects on air temperature due to water vapor although we will still calculate pressure changes adiabatically using our adjusted value of \(\lambda\). Any water vapor will likely have condensed by the time airflow begins. Thrust from choked airflow When the ratio of ambient pressure to total absolute tank pressure is less than the "choke ratio"
$$\alpha_c = \frac{P_a}{P_e} = \left(\frac{2}{\lambda+1}\right)^\frac{\lambda}{\lambda-1} \tag{A1}$$
then the outflow is choked, or limited, to the speed of sound:
$$c = \sqrt{\lambda RT} \tag{A2}$$
where\(c\) = speed of sound in air, approximately 343 m/s at 20°C, 331 m/s at 0°C\(R\) = ideal…

Once the rocket has left the launch tube, the water thrust phase of the flight begins. But before we begin those calculations, we need to have equations for ballistic flight, which is the last step described in the introduction. Why? Because during ballistic flight (coasting through the air), the only forces acting on the rocket are gravity and wind drag — but these are acting on the rocket during the thrust phase too. Ballistic flight is exactly the same, just without the thrust. Ballistic flight We need to know how gravity and drag affect acceleration, velocity, and ultimately altitude during all thrust phases as well as the ballistic trajectory afterward, so we'll start with the ballistic flight equations.Air resistanceFirst we need to get the ambient air density. Denser air results in higher drag, and dry air is more dense than moist air. To get the density, we first need to know the partial pressure of water vapor in the air. To do this, we first get the saturation vapor pres…

A launch tube inside the bottle does two important things:A tube with its open end above the water level keeps the water from spilling into the launcher pipe-work, leaving it inside the bottle for the rocket to use as reaction mass.It allows the bottle to gain velocity as the nozzle slides along the length of the tube, with negligible leakage of the water reaction mass.There are a large number of international standards for 28 mm soda bottle necks. All the PET soda bottles I've seen look like PCO-1881 (PDF) evidenced by the slope of the flange that holds the bottle cap's retaining ring. In any case, all these standards have the same inner diameter: 21.74 mm. Let's assume 21.75 mm for wear and tear, as well as conservatism when calculating leakage. A standard ½-inch Schedule 40 PVC pipe has an outside diameter of 0.840” or 21.36 mm, which fits nicely into the bottle neck with about a 0.19 mm gap all around. The force acting on the bottle is simply the internal pressure mult…

I recently became interested in water rocketry after realizing how complicated the physics actually can be, and the fact that the equations are best solved with numerical methods because so many interacting variables change rapidly in a non-linear fashion. For those unfamiliar, a water rocket is a rocket that uses water as its reaction mass, powered by air pressure. The cool thing about it is the easy availability of pressure vessels — soda bottles. You get a free rocket body with each soft drink purchase! A typical soda bottle can withstand an internal pressure of 100 psi easily, and can go up to 160 psi before rupturing.Here's the basic bare-minimum set-up, without explaining how the launcher works. There are plenty of tutorials on building a launcher.Launching just a bottle without anything else attached.Source: by RadioActive~commonswiki on Wikimedia Commons It sounds like a low-cost hobby, right? Wrong. Well, sort of. Cheap is possible if you just want to have fun and aren'…

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Once the rocket has left the launch tube, the water thrust phase of the flight begins. But before we begin those calculations, we need to have equations for ballistic flight, which is the last step described in the introduction. Why? Because during ballistic flight (coasting through the air), the only forces acting on the rocket are gravity and wind drag — but these are acting on the rocket during the thrust phase too. Ballistic flight is exactly the same, just without the thrust. Ballistic flight We need to know how gravity and drag affect acceleration, velocity, and ultimately altitude during all thrust phases as well as the ballistic trajectory afterward, so we'll start with the ballistic flight equations.Air resistanceFirst we need to get the ambient air density. Denser air results in higher drag, and dry air is more dense than moist air. To get the density, we first need to know the partial pressure of water vapor in the air. To do this, we first get the saturation vapor pres…

A launch tube inside the bottle does two important things:A tube with its open end above the water level keeps the water from spilling into the launcher pipe-work, leaving it inside the bottle for the rocket to use as reaction mass.It allows the bottle to gain velocity as the nozzle slides along the length of the tube, with negligible leakage of the water reaction mass.There are a large number of international standards for 28 mm soda bottle necks. All the PET soda bottles I've seen look like PCO-1881 (PDF) evidenced by the slope of the flange that holds the bottle cap's retaining ring. In any case, all these standards have the same inner diameter: 21.74 mm. Let's assume 21.75 mm for wear and tear, as well as conservatism when calculating leakage. A standard ½-inch Schedule 40 PVC pipe has an outside diameter of 0.840” or 21.36 mm, which fits nicely into the bottle neck with about a 0.19 mm gap all around. The force acting on the bottle is simply the internal pressure mult…

So now we're ready to put all the calculations together: launch tube, water thrust, air thrust, and ballistic flight. The launch tube calculation is more like an initialization for the water thrust phase; we just have one time step of the duration of launch tube traversal. Get the initial pressure \(P_0\) for water thrust from (L1), and the velocity \(v_0\) from (L4), and use these values in the first water thrust time step. You may also want to initialize the altitude achieved as the height of the tube. Water thrust calculationFor the water thrust phase, we do the following calculations at each time interval, which should be 1 millisecond or less in duration:At the beginning of time interval \(i\):Get the pressure \(P_1\) from (W4), using the change in water volume from the previous time interval.From the water volume, calculate the height of the water in the bottle — needed for \(H\) in equation (W1). This can be approximated by calculating the height of the water in a cylinder …